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Mirrors > Home > ILE Home > Th. List > nqpru | Unicode version |
Description: Comparing a fraction to a real can be done by whether it is an element of the upper cut, or by . (Contributed by Jim Kingdon, 29-Nov-2020.) |
Ref | Expression |
---|---|
nqpru |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prop 6573 | . . . . . 6 | |
2 | prnminu 6587 | . . . . . 6 | |
3 | 1, 2 | sylan 267 | . . . . 5 |
4 | elprnqu 6580 | . . . . . . . . . 10 | |
5 | 1, 4 | sylan 267 | . . . . . . . . 9 |
6 | 5 | ad2ant2r 478 | . . . . . . . 8 |
7 | simprl 483 | . . . . . . . 8 | |
8 | vex 2560 | . . . . . . . . . . . 12 | |
9 | breq1 3767 | . . . . . . . . . . . 12 | |
10 | 8, 9 | elab 2687 | . . . . . . . . . . 11 |
11 | 10 | biimpri 124 | . . . . . . . . . 10 |
12 | ltnqex 6647 | . . . . . . . . . . . 12 | |
13 | gtnqex 6648 | . . . . . . . . . . . 12 | |
14 | 12, 13 | op1st 5773 | . . . . . . . . . . 11 |
15 | 14 | eleq2i 2104 | . . . . . . . . . 10 |
16 | 11, 15 | sylibr 137 | . . . . . . . . 9 |
17 | 16 | ad2antll 460 | . . . . . . . 8 |
18 | 19.8a 1482 | . . . . . . . 8 | |
19 | 6, 7, 17, 18 | syl12anc 1133 | . . . . . . 7 |
20 | df-rex 2312 | . . . . . . 7 | |
21 | 19, 20 | sylibr 137 | . . . . . 6 |
22 | elprnqu 6580 | . . . . . . . . 9 | |
23 | 1, 22 | sylan 267 | . . . . . . . 8 |
24 | nqprlu 6645 | . . . . . . . . 9 | |
25 | ltdfpr 6604 | . . . . . . . . 9 | |
26 | 24, 25 | sylan2 270 | . . . . . . . 8 |
27 | 23, 26 | syldan 266 | . . . . . . 7 |
28 | 27 | adantr 261 | . . . . . 6 |
29 | 21, 28 | mpbird 156 | . . . . 5 |
30 | 3, 29 | rexlimddv 2437 | . . . 4 |
31 | 30 | ex 108 | . . 3 |
32 | 31 | adantl 262 | . 2 |
33 | 26 | ancoms 255 | . . . . 5 |
34 | 33 | biimpa 280 | . . . 4 |
35 | 15, 10 | bitri 173 | . . . . . . . 8 |
36 | 35 | biimpi 113 | . . . . . . 7 |
37 | 36 | ad2antll 460 | . . . . . 6 |
38 | 37 | adantl 262 | . . . . 5 |
39 | simpllr 486 | . . . . . 6 | |
40 | simprrl 491 | . . . . . 6 | |
41 | prcunqu 6583 | . . . . . . 7 | |
42 | 1, 41 | sylan 267 | . . . . . 6 |
43 | 39, 40, 42 | syl2anc 391 | . . . . 5 |
44 | 38, 43 | mpd 13 | . . . 4 |
45 | 34, 44 | rexlimddv 2437 | . . 3 |
46 | 45 | ex 108 | . 2 |
47 | 32, 46 | impbid 120 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wex 1381 wcel 1393 cab 2026 wrex 2307 cop 3378 class class class wbr 3764 cfv 4902 c1st 5765 c2nd 5766 cnq 6378 cltq 6383 cnp 6389 cltp 6393 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-1o 6001 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-lti 6405 df-plpq 6442 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-mqqs 6448 df-1nqqs 6449 df-rq 6450 df-ltnqqs 6451 df-inp 6564 df-iltp 6568 |
This theorem is referenced by: prplnqu 6718 caucvgprprlemmu 6793 caucvgprprlemopu 6797 caucvgprprlemexbt 6804 caucvgprprlem2 6808 |
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